"You are disoriented. Blackness swims toward you like a school of eels who have just seen something that eels like a lot." ~ Douglas Adams

Why do roots come in conjugate pairs?

This is an expanded version of an answer I gave to a question that came up while I was assisting the 2014-2015 WOOT class. It struck me as an unusually good way to motivate higher math using stuff that people notice in high school but for some reason decide to not think about.

In high school precalculus, you’ll often be asked to find the roots of some polynomial with integer coefficients. For instance,

$latex displaystyle x^3 – x^2 – x – 15 = (x-3)(x^2+2x=5) &fg=000000$

has roots $latex {3}&fg=000000$, $latex {1+2i}&fg=000000$, $latex {-1-2i}&fg=000000$. Or as another example,

"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." - Paul Halmos